I assume you will get a car without registration for public roads in order to avoid the insane Danish taxes, right ?
If so, you indeed may look for a "hardcore solution", though I think a Radical (even a used one) might not fit into your budget
The Cup Clios are really fast (and I'm sure they are fun to drive also), though you might ask some owners about reliability/maintenance costs etc.
I'd put the Elise on number one of my short-list for the reasons already mentioned by Gnil
It's a good track performer out of the box - though the Ring-maniacs claim, that the suspension is perfect for NoS, but could be a bit "sharper" for a real (smooth) track.
Also maintenance seems to be not big $$$ (some Ring-maniacs do 500 laps/10k km p.a. without any extraordinary expenses other than brake pads/tires).
The Opel Speedster would be my budget solution (definitely less money compared to the Elise, but not that focussed - e.g. heavier than Elise).
I guess the Renault Spider might be harder to find (seems to be more a collector's item).
Let me know your decision
I have started to look myself (will let you know what I'll find out during the search process)